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# Polytope of Type {4,4,3}

Atlas Canonical Name : {4,4,3}*192b
Also Known As : 1T4(2,0), {{4,4|2},{4,3}}. if this polytope has another name.
Group : SmallGroup(192,1472)
Rank : 4
Schlafli Type : {4,4,3}
Number of vertices, edges, etc : 4, 16, 12, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,3,2} of size 384
{4,4,3,3} of size 768
{4,4,3,4} of size 768
{4,4,3,6} of size 1152
Vertex Figure Of :
{2,4,4,3} of size 384
{4,4,4,3} of size 768
{6,4,4,3} of size 1152
{3,4,4,3} of size 1152
{6,4,4,3} of size 1728
{10,4,4,3} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,3}*96
4-fold quotients : {4,2,3}*48, {2,4,3}*48
8-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8,3}*384, {8,4,3}*384, {4,4,6}*384d
3-fold covers : {4,4,9}*576b, {12,4,3}*576, {4,12,3}*576
4-fold covers : {8,8,3}*768, {4,4,3}*768a, {4,8,3}*768c, {4,8,3}*768d, {16,4,3}*768, {4,4,6}*768e, {4,4,12}*768e, {4,4,12}*768f, {4,8,6}*768c, {8,4,6}*768c, {4,8,6}*768d
5-fold covers : {20,4,3}*960, {4,4,15}*960b
6-fold covers : {4,8,9}*1152, {8,4,9}*1152, {4,4,18}*1152d, {12,8,3}*1152, {24,4,3}*1152, {8,12,3}*1152, {4,24,3}*1152, {12,4,6}*1152c, {4,12,6}*1152g, {4,12,6}*1152j
7-fold covers : {28,4,3}*1344, {4,4,21}*1344b
9-fold covers : {4,4,27}*1728b, {36,4,3}*1728, {12,4,9}*1728, {12,12,3}*1728a, {4,12,9}*1728, {4,12,3}*1728a, {12,12,3}*1728b, {4,12,3}*1728b
10-fold covers : {20,8,3}*1920, {40,4,3}*1920, {4,8,15}*1920, {8,4,15}*1920, {20,4,6}*1920b, {4,20,6}*1920c, {4,4,30}*1920d
Permutation Representation (GAP) :
```s0 := ( 7, 8)( 9,10)(11,12);;
s1 := ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,12)( 6,11);;
s2 := ( 3, 5)( 4, 6)( 9,11)(10,12);;
s3 := ( 1, 3)( 2, 4)( 7, 9)( 8,10);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!( 7, 8)( 9,10)(11,12);
s1 := Sym(12)!( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,12)( 6,11);
s2 := Sym(12)!( 3, 5)( 4, 6)( 9,11)(10,12);
s3 := Sym(12)!( 1, 3)( 2, 4)( 7, 9)( 8,10);
poly := sub<Sym(12)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References :
1. Theorem 10B3, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

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